Asymptotic diophantine approximation: the multiplicative case

Let αα and ββ be irrational real numbers and 0<ε<1/300<ε<1/30. We prove a precise estimate for the number of positive integers q≤Qq≤Q that satisfy ∥qα∥⋅∥qβ∥<ε∥qα∥⋅∥qβ∥<ε. If we choose εε as a function of Q, we get asymptotics as Q gets large, provided εQεQ grows quickly enough in terms of the (multiplicative) Diophantine type of (α,β)(α,β), e.g., if (α,β)(α,β) is a counterexample to Littlewood’s conjecture, then we only need that εQεQ tends to infinity. Our result yields a new upper bound on sums of reciprocals of products of fractional parts and sheds some light on a recent question of Lê and Vaaler.